How many prime numbers between 30 and 65 have a prime remainder when divided by 10?
Solution: If a prime number between 30 and 65 has a prime remainder when divided by 10, it must end in 3 or 7, since any number greater than 10 ending in 2 or 5 is composite.  Systemically checking all possibilities, we see that the only such primes are 37, 43, 47, and 53, which gives us a total of $\boxed{4}$ such numbers.